Just to clarify the issue: For example, the predictions based on log
(E[price]) = XG with GLM should be identical to the predictions
generated from E[log(price)] = XB (fit by -regress-, generating
B_hat), when the later are adjusted properly?

I'm not sure what you mean by "adjusted" here, but no, the
predictions will not be the same.

What would you suggest for predictions based on a box-cox (left-
hand side) transformation? A two step procedure, first estimating
the box-cox transformation parameter and then using that parameter
in a GLM to generate predicted variables?

No, I would not suggest that. These are two different approaches to
the problem. In one case, you're assuming that a single
transformation will address both the mean structure of the model and
the error structure. In the second (GLM), you're modeling the error
structure on the raw (untransformed) scale, and using the link
function merely to specify the relationship between the mean of the
response and the linear predictor (i.e., XB). So no, there's no
reason in general to expect that an appropriate transformation in the
first case will correspond to an appropriate link function (or vice
versa).

I don't mean to be rude, but if you want to understand the GLM
approach, you'll need to spend some time with one of the many books
which explain it. If, after that, you have a specific question about
something you have read (or, better yet, how something you read
translates into Stata usage), then many on this list would, I'm sure,
be glad to try to answer it.